1. Field of the Invention
This application relates to Discrete Fourier Transforms, and in particular, to Discrete Fourier Transforms in phase retrieval for use in waveform analysis.
2. Background
Phase retrieval is an image-based wavefront sensing method that utilizes point-source images (or other known objects) to recover optical phase information. The most famous application of phase retrieval was the diagnosis of the Hubble Space Telescope mirror edge defect discovered soon after the launch of Hubble and subsequent correction using Corrective Optics Space Telescope Axial Replacement (COSTAR.) It may be ironic that a phase retrieval wavefront-sensing method lies at the very heart of the commissioning process for Hubble's successor. The earliest suggestion of using phase retrieval as a wavefront sensing method for JWST was in 1989, nearly a year before the Hubble launch and deployment. Details documenting the Hubble Space Telescope phase retrieval analysis are discussed in the literature.
A number of image-based phase retrieval techniques have been developed that can be classified into two general categories, the (a) iterative-transform and (b) parametric methods. Modifications to the original iterative-transform approach have been based on the introduction of a defocus diversity function or on the input-output method. Various implementations of the focus-diverse iterative-transform method have appeared which deviate slightly by utilizing a single wavelength or by varying the placement and number of defocused image planes. Modifications to the parametric approach include minimizing alternative merit functions as well as implementing a variety of nonlinear optimization methods such as Levenburg-Marquardt, simplex, and quasi-Newton techniques. The concept behind an optical diversity function is to modulate a point source image in a controlled way; in principle, any known aberration can serve as a diversity function, but defocus is often the simplest to implement and exhibits no angular dependence as a function of the pupil coordinates
The discrete Fourier transform (DFT) calculation is presented in “diagonal” form. By diagonal we mean that a transformation of basis is introduced by an application of the similarity transform of linear algebra.
What is needed is to find a more efficient implementation of the DFT for applications using iterative transform methods, particularly phase retrieval.